A333284 - OEIS

%I #29 May 21 2021 07:15:40

%S 5,13,37,35,129,405,75,289,933,2225,159,663,2155,5157,11641,275,1163,

%T 3793,9051,20341,35677,477,2069,6771,16129,36173,63987,114409,755,

%U 3251,10727,25635,57759,102845,183961

%N Triangle read by rows: T(m,n) (m >= n >= 1) = number of vertices formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

%C If we only joined pairs of the 2(m+n) boundary points, we would get A331453. If we did not extend the lines to the boundary of the grid, we would get A288180. (One of the links below shows the difference between the three definitions in the case m=3, n=2.)

%C See A333282 for a large number of colored illustrations.

%H Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, <a href="http://neilsloane.com/doc/rose_5.pdf">Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids</a>, (2021). Also arXiv:2009.07918.

%H Seppo Mustonen, <a href="http://www.survo.fi/papers/GeomAccuracy.pdf">Statistical accuracy of geometric constructions</a>, 2008.

%H Seppo Mustonen, <a href="/A333282/a333282_1.pdf">Statistical accuracy of geometric constructions</a>, 2008 [Local copy]

%H Seppo Mustonen, <a href="http://www.survo.fi/papers/PointsInGrid.pdf">On lines and their intersection points in a rectangular grid of points</a>, 2009

%H Seppo Mustonen, <a href="/A018808/a018808.pdf">On lines and their intersection points in a rectangular grid of points</a>, 2009 [Local copy]

%H Seppo Mustonen, <a href="http://www.survo.fi/papers/LinesInGrid2.pdf">On lines going through a given number of points in a rectangular grid of points</a>, 2010

%H Seppo Mustonen, <a href="/A141255/a141255.pdf">On lines going through a given number of points in a rectangular grid of points</a>, 2010 [Local copy]

%H N. J. A. Sloane, <a href="/A333282/a333282.pdf">Illustration of T(3,2) = 129.</a> [Black lines correspond to A331453(3,2), black + red lines correspond to A288180(3,2), and black + red + blue lines to T(3,2)]

%H N. J. A. Sloane, <a href="/A333282/a333282_2.pdf">Illustration of T(3,3) = 405</a> [Black lines correspond to A288180(3,3), and black + red lines to T(3,3)]

%e Triangle begins:

%e 5,

%e 13, 37,

%e 35, 129, 405,

%e 75, 289, 933, 2225,

%e 159, 663, 2155, 5157, 11641,

%e 275, 1163, 3793, 9051, 20341, 35677,

%e 477, 2069, 6771, 16129, 36173, 63987, 114409,

%e 755, 3251, 10727, 25635, 57759, 102845, 183961, ...

%e ...

%e T(7,7) corrected Mar 19 2020

%Y Cf. A288187, A331452, A288180, A331453, A333282 (regions), A333283 (edges). Column 1 is A331755. The main diagonal is A333285.

%K nonn,tabl,more

%O 1,1

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 16 2020

%E More terms and corrections from _Scott R. Shannon_, Mar 21 2020